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Theorem rexuz2 11210
Description: Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
Assertion
Ref Expression
rexuz2  |-  ( E. n  e.  ( ZZ>= `  M ) ph  <->  ( M  e.  ZZ  /\  E. n  e.  ZZ  ( M  <_  n  /\  ph ) ) )
Distinct variable group:    n, M
Allowed substitution hint:    ph( n)

Proof of Theorem rexuz2
StepHypRef Expression
1 eluz2 11165 . . . . . 6  |-  ( n  e.  ( ZZ>= `  M
)  <->  ( M  e.  ZZ  /\  n  e.  ZZ  /\  M  <_  n ) )
2 df-3an 984 . . . . . 6  |-  ( ( M  e.  ZZ  /\  n  e.  ZZ  /\  M  <_  n )  <->  ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  M  <_  n ) )
31, 2bitri 252 . . . . 5  |-  ( n  e.  ( ZZ>= `  M
)  <->  ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  M  <_  n ) )
43anbi1i 699 . . . 4  |-  ( ( n  e.  ( ZZ>= `  M )  /\  ph ) 
<->  ( ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  M  <_  n )  /\  ph ) )
5 anass 653 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  M  <_  n )  /\  ph ) 
<->  ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  ( M  <_  n  /\  ph ) ) )
6 anass 653 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  ( M  <_  n  /\  ph ) )  <-> 
( M  e.  ZZ  /\  ( n  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
7 an12 804 . . . . . 6  |-  ( ( M  e.  ZZ  /\  ( n  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) )  <-> 
( n  e.  ZZ  /\  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
86, 7bitri 252 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  ( M  <_  n  /\  ph ) )  <-> 
( n  e.  ZZ  /\  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
95, 8bitri 252 . . . 4  |-  ( ( ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  M  <_  n )  /\  ph ) 
<->  ( n  e.  ZZ  /\  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
104, 9bitri 252 . . 3  |-  ( ( n  e.  ( ZZ>= `  M )  /\  ph ) 
<->  ( n  e.  ZZ  /\  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
1110rexbii2 2932 . 2  |-  ( E. n  e.  ( ZZ>= `  M ) ph  <->  E. n  e.  ZZ  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) )
12 r19.42v 2990 . 2  |-  ( E. n  e.  ZZ  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) )  <->  ( M  e.  ZZ  /\  E. n  e.  ZZ  ( M  <_  n  /\  ph ) ) )
1311, 12bitri 252 1  |-  ( E. n  e.  ( ZZ>= `  M ) ph  <->  ( M  e.  ZZ  /\  E. n  e.  ZZ  ( M  <_  n  /\  ph ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    /\ w3a 982    e. wcel 1870   E.wrex 2783   class class class wbr 4426   ` cfv 5601    <_ cle 9675   ZZcz 10937   ZZ>=cuz 11159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-cnex 9594  ax-resscn 9595
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-neg 9862  df-z 10938  df-uz 11160
This theorem is referenced by:  2rexuz  11211
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